Question

The single proton that forms the nucleus of the hydrogen atom has a radius of approximately \(1.0 \times 10^{-13} \mathrm{~cm} .\) The hydrogen atom itself has a radius of approximately \(52.9 \mathrm{pm} .\) What fraction of the space within the atom is occupied by the nucleus?

Short answer

The fraction of space occupied by the nucleus is approximately \(4.8 \times 10^{-5}\).

Step-by-step solution

Convert the radius of the hydrogen atom to centimeters

Since the radius of the hydrogen nucleus is given in centimeters, we need to convert the radius of the hydrogen atom from picometers (pm) to centimeters for consistency. There are 1x10^12 picometers in a centimeter. Therefore, to convert from picometers to centimeters, divide the number of picometers by 1x10^12.

Calculate the volume of the nucleus

Use the formula for the volume of a sphere, which is \(V = \frac{4}{3} \pi r^3\), where \(r\) is the radius of the sphere. Calculate the volume of the nucleus with the given radius in centimeters.

Calculate the volume of the hydrogen atom

Now that we have the radius of the hydrogen atom in centimeters, calculate its volume using the same formula for the volume of a sphere.

Calculate the fraction of space occupied by the nucleus

To find the fraction of the space within the atom that is occupied by the nucleus, divide the volume of the nucleus by the volume of the atom.

Key concepts

Nuclear Radius

The term 'nuclear radius' refers to the measure of the size of an atomic nucleus and is typically on the order of 1 femtometer (1 fm), which is equivalent to 1x10^-15 meters. To put this in perspective, if an atom were the size of a football stadium, the nucleus would be only the size of a pea at the center of the field. Understanding the nuclear radius is crucial in fields such as nuclear physics and chemistry, as it impacts the behavior and interaction of atoms.

In our exercise, the nuclear radius of a hydrogen atom's proton is given as approximately 1.0x10^-13 cm. While this may seem tiny, it is immensely dense and contains virtually all of an atom's mass. This microscopic scale is challenging to visualize, but it's fundamental to conceptualizing the atom's structure and the vast 'empty' space it encompasses.

Atomic Radius

In contrast to the nuclear radius, the atomic radius represents the distance from the center of the nucleus to the outer boundary of the surrounding cloud of electrons. Atomic radius is typically measured in picometers (pm), where 1 pm equals 1x10^-12 meters, and the values vary for different elements on the periodic table. For instance, the atomic radius of hydrogen is about 52.9 pm, indicating where its electron cloud extends.

The atomic radius provides insight into many properties of an element, including its reactivity and bonding capabilities. As atoms bond, the space their electron clouds occupy dictates the type and strength of the bonds that can form. Consequently, knowing an atom's radius is critical for understanding and predicting chemical behavior.

Volume of a Sphere

The volume of a sphere is given by the mathematical formula \(V = \frac{4}{3} \pi r^3\), where \(r\) represents the radius of the sphere. This formula allows us to compute how much three-dimensional space the sphere occupies. It's crucial in many scientific calculations, like determining the space within an atom that's filled by the nucleus or the electron cloud.

In our exercise, this formula is applied twice: once to determine the volume of the nucleus using the nuclear radius and then to find the volume of the entire atom using its atomic radius. These calculations enable us to ultimately determine what fraction of the total volume of a hydrogen atom is actually occupied by its nucleus.

Picometers to Centimeters Conversion

Converting measurements from picometers to centimeters is essential when dealing with atomic and subatomic scales, as it allows us to express dimensions in a consistent unit of measure. Since 1 picometer is 1x10^-12 meters, and 1 centimeter is 1x10^-2 meters, we use the conversion factor of 1x10¹⁰ to go from picometers to centimeters (divide the number of picometers by 1x10¹² to get centimeters).

In the context of our problem, converting the atomic radius of a hydrogen atom from 52.9 pm to centimeters is a necessary step. Without this conversion, we wouldn't be able to directly compare the size of the nucleus to that of the atom since we would be dealing with different units of measurement. This conversion is a fundamental skill in scientific calculations because it ensures that all measurements are in the same unit, which is crucial for accurate computations and comparisons.